deffunc H_{1}( Element of X, Element of X, Element of Y, Element of Y) -> Element of REAL = In ((sqrt (((dist ($1,$2)) ^2) + ((dist ($3,$4)) ^2))),REAL);

consider F being Function of [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],REAL such that

A1: for x1, y1 being Element of X

for x2, y2 being Element of Y

for x, y being Element of [: the carrier of X, the carrier of Y:] st x = [x1,x2] & y = [y1,y2] holds

F . (x,y) = H_{1}(x1,y1,x2,y2)
from METRIC_3:sch 1();

take F ; :: thesis: for x1, y1 being Element of X

for x2, y2 being Element of Y

for x, y being Element of [: the carrier of X, the carrier of Y:] st x = [x1,x2] & y = [y1,y2] holds

F . (x,y) = sqrt (((dist (x1,y1)) ^2) + ((dist (x2,y2)) ^2))

let x1, y1 be Element of X; :: thesis: for x2, y2 being Element of Y

for x, y being Element of [: the carrier of X, the carrier of Y:] st x = [x1,x2] & y = [y1,y2] holds

F . (x,y) = sqrt (((dist (x1,y1)) ^2) + ((dist (x2,y2)) ^2))

let x2, y2 be Element of Y; :: thesis: for x, y being Element of [: the carrier of X, the carrier of Y:] st x = [x1,x2] & y = [y1,y2] holds

F . (x,y) = sqrt (((dist (x1,y1)) ^2) + ((dist (x2,y2)) ^2))

let x, y be Element of [: the carrier of X, the carrier of Y:]; :: thesis: ( x = [x1,x2] & y = [y1,y2] implies F . (x,y) = sqrt (((dist (x1,y1)) ^2) + ((dist (x2,y2)) ^2)) )

assume ( x = [x1,x2] & y = [y1,y2] ) ; :: thesis: F . (x,y) = sqrt (((dist (x1,y1)) ^2) + ((dist (x2,y2)) ^2))

then F . (x,y) = H_{1}(x1,y1,x2,y2)
by A1;

hence F . (x,y) = sqrt (((dist (x1,y1)) ^2) + ((dist (x2,y2)) ^2)) ; :: thesis: verum

consider F being Function of [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],REAL such that

A1: for x1, y1 being Element of X

for x2, y2 being Element of Y

for x, y being Element of [: the carrier of X, the carrier of Y:] st x = [x1,x2] & y = [y1,y2] holds

F . (x,y) = H

take F ; :: thesis: for x1, y1 being Element of X

for x2, y2 being Element of Y

for x, y being Element of [: the carrier of X, the carrier of Y:] st x = [x1,x2] & y = [y1,y2] holds

F . (x,y) = sqrt (((dist (x1,y1)) ^2) + ((dist (x2,y2)) ^2))

let x1, y1 be Element of X; :: thesis: for x2, y2 being Element of Y

for x, y being Element of [: the carrier of X, the carrier of Y:] st x = [x1,x2] & y = [y1,y2] holds

F . (x,y) = sqrt (((dist (x1,y1)) ^2) + ((dist (x2,y2)) ^2))

let x2, y2 be Element of Y; :: thesis: for x, y being Element of [: the carrier of X, the carrier of Y:] st x = [x1,x2] & y = [y1,y2] holds

F . (x,y) = sqrt (((dist (x1,y1)) ^2) + ((dist (x2,y2)) ^2))

let x, y be Element of [: the carrier of X, the carrier of Y:]; :: thesis: ( x = [x1,x2] & y = [y1,y2] implies F . (x,y) = sqrt (((dist (x1,y1)) ^2) + ((dist (x2,y2)) ^2)) )

assume ( x = [x1,x2] & y = [y1,y2] ) ; :: thesis: F . (x,y) = sqrt (((dist (x1,y1)) ^2) + ((dist (x2,y2)) ^2))

then F . (x,y) = H

hence F . (x,y) = sqrt (((dist (x1,y1)) ^2) + ((dist (x2,y2)) ^2)) ; :: thesis: verum